The tutorial on “Principal Component Analysis (PCA) in R” by: Statistics Globe
R ENVIRONMENT SET UP & DATA
Needed R Packages
We will use functions from packages base, utils, and stats (pre-installed and pre-loaded)
We may also use the packages below (specifying package::function for clarity).
# Load pckgs for this R sessionoptions(scipen =999)# --- General library(here) # tools find your project's files, based on working directorylibrary(dplyr) # A Grammar of Data Manipulationlibrary(skimr) # Compact and Flexible Summaries of Datalibrary(magrittr) # A Forward-Pipe Operator for R library(readr) # A Forward-Pipe Operator for R library(tidyr) # Tidy Messy Datalibrary(kableExtra) # Construct Complex Table with 'kable' and Pipe Syntax# ---Plotting & data visualizationlibrary(ggplot2) # Create Elegant Data Visualisations Using the Grammar of Graphicslibrary(ggfortify) # Data Visualization Tools for Statistical Analysis Resultslibrary(scatterplot3d) # 3D Scatter Plot# --- Statisticslibrary(MASS) # Support Functions and Datasets for Venables and Ripley's MASSlibrary(factoextra) # Extract and Visualize the Results of Multivariate Data Analyseslibrary(FactoMineR) # Multivariate Exploratory Data Analysis and Data Mininglibrary(rstatix) # Pipe-Friendly Framework for Basic Statistical Testslibrary(car) # Companion to Applied Regressionlibrary(ROCR) # Visualizing the Performance of Scoring Classifiers# --- Tidymodels (meta package)library(rsample) # General Resampling Infrastructure library(broom) # Convert Statistical Objects into Tidy Tibbles
LOGISTIC REGRESSION
(EXAMPLE of SUPERVISED ML ALGORITHM)
Logistic regression: review
Logistic regression is a classification model used with a binary response variable, e.g.:
yes|no, or 0|1, or True|False in a survey question;
success|failure in a clinical trial experiment;
benign|malignant in a biopsy experiment.
Logistic regression is a type of Generalized Linear Model (GLM): a more flexible version of linear regression that can work also for categorical response variables or count data (e.g. poisson regression).
When logistic regression fits the coefficients \(\beta_0\), \(\beta_1\), …, \(\beta_k\) to the data, it minimizes errors using the Maximum Likelihood Estimation method (as opposed to linear regression’s Least Squares Error Estimation method).
Logistic regression: logit function
If we have predictor variables like \(x_{1,i}\), \(x_{2,i}\), …, \(x_{k,i}\) and a binary response variable \(y_i\) (where \(y_i = 0\) or \(y_i = 1\)), we need a “special” function to transform the expected value of the response variable into the \([0,1]\) outcome we’re trying to predict.
The logit function (of the GLM family) helps us determine the coefficients \(\beta_0\), \(\beta_1\), …, \(\beta_k\) that best fit this sort of data and it is defined as: \[
logit (p_i) = \ln\left( \frac{p_i}{1-p_i} \right)= \beta_0 + \beta_1 x_{1,i} + \cdots + \beta_k x_{k,i}
\] where \(p_i\) is the probability that \(y_i = 1\)
Via the inverse-logit function (logistic), we solve for the probability \(p_i\), given values of the predictor variables, like so:
a few clean datasets used in the “Core Statistics using R” course by: Martin van Rongen
Dataset on Heart Disease (heart_data)
Name: heart_data.csv Documentation: Toy dataset prepared for teaching purposes. See reference on the data here Data Analytics with R Sampling details: This dataset contains 10,000 observations on 4 variables.
# Use `here` in specifying all the subfolders AFTER the working directory heart_data <-read.csv(file = here::here("practice", "data_input", "05_datasets","heart_data.csv"), header =TRUE, # 1st line is the name of the variablessep =",", # which is the field separator character.na.strings =c("?","NA" ), # specific MISSING values row.names =NULL)
heart_data variables with description
Variable
Type
Description
heart_disease
int
whether an individual has heart disease (1 = yes; 0 = no)
a numerical field corresponding to the annual spend of each individual on fast food
income
dbl
a numerical field corresponding to the individual’s annual income
heart_data dataset splitting
In Machine Learning, it is good practice to split the data into training and testing sets.
We will use the training set (70%) to fit the model and then the testing set (30%) to evaluate the model’s performance.
set.seed(123)# Obtain 2 sub-samples from the dataset: training and testingsample <-sample(c(TRUE, FALSE), nrow(heart_data), replace =TRUE , prob =c(0.7, 0.3) )heart_train <- heart_data[sample,]heart_test <- heart_data[!sample,]
Which results in:
# check the structure of the resulting datasetsdim(heart_train)
[1] 7048 4
dim(heart_test)
[1] 2952 4
Convert binary variables to factors
Before examining the training dataset heart_train, we converting the binary variables heart_disease and coffee_drinker to factors (for better readability).
heart_train <- heart_train %>%# convert to factor with levels "Yes" and "No"mutate(heart_disease =factor(heart_disease, levels =c(0, 1),labels =c("No_HD", "Yes_HD")),coffee_drinker =factor(coffee_drinker, levels =c(0, 1),labels =c("No_Coffee", "Yes_Coffee")) )# show the first 5 rows of the datasetheart_train[1:5,]
We now examine visualize the relationship between the binary outcome variable heart_disease and the continuous predictor variable fast_food_spend.
geom_jitter adds a bit of randomness to the points to avoid overplotting.
geom_boxplot shows the distribution of the continuous variable by the binary outcome variable.
# plot the distribution of heart disease status by fast food spendheart_train %>%ggplot(aes(x = heart_disease, y = fast_food_spend, fill = heart_disease)) +geom_jitter(aes(fill = heart_disease), alpha =0.3, shape =21, width =0.25) +scale_color_manual(values =c("#005ca1", "#9b2339")) +scale_fill_manual(values =c("#57b7ff", "#e07689")) +geom_boxplot(fill =NA, color ="black", linewidth = .7) +coord_flip() +theme(plot.title =element_text(size =13,face="bold", color ="#873c4a"),axis.text.x =element_text(size=12,face="italic"), axis.text.y =element_text(size=12,face="italic"),legend.position ="none") +labs(title ="Fast food expenditure by heart disease status") +xlab("Heart Disease (Y/N)") +ylab("Annual Fast Food Spend")
Plotting Y by X1 (continuous variable)
+ The boxplots indicate that subjects with heart disease (HD =1) seem to spend higher amounts on fast food + Also, this sample has many more subjects without heart disease (HD = 0) than with heart disease (HD = 1)
Plotting Y by X2 (discrete variable)
Then we examine the relationship between the binary outcome variable heart_disease and the binary predictor variable coffee_drinker.
we use the handy count function from dplyr to count occurrences in categorical variable(s) combinations.
# Dataset manipulationheart_train %>%# count the unique values per each group from 2 categorical variables' combinations dplyr::count(heart_disease, coffee_drinker, name ="count_by_group") %>% dplyr::group_by(coffee_drinker) %>% dplyr::mutate(total_coffee_class =sum(count_by_group),proportion = count_by_group / total_coffee_class) %>% dplyr::ungroup() %>%# filter only those with heart disease dplyr::filter(heart_disease =="Yes_HD") %>%# Plot ggplot(aes(x = coffee_drinker, y = proportion, fill = coffee_drinker)) +geom_bar(stat ="identity") +scale_fill_manual(values =c("#57b7ff", "#e07689")) +theme_minimal() +ylab("Percent with Heart Disease") +xlab("Coffee Drinker (Y/N)") +ggtitle("Figure 3: Percent of Coffee Drinkers with Heart Disease") +labs(fill ="Coffee Drinker") +scale_y_continuous(labels = scales::percent)
Plotting Y by X2 (discrete variable)
Also drinking coffee seems associated to a higher likelihood of heart disease (HD =1)
Linear regression wouldn’t work!
In principle, we could use a linear regression model to study likelihood of having Heart Disease in relation to risk factors: \[ Y = \beta_0 + \beta_1 \cdot \text{(US\$ Spent Fast Food)} + \beta_2 \cdot \text{(Coffee Drinker = YES)} \]
But we’ll see why the logistic regression model is a better option.
Let’s compare the 2 models (for simplicity, we ignore the coffee_drinker variable for now.):
# --- 1) Linear regression modellinear_mod <-lm(heart_disease ~ fast_food_spend# + coffee_drinker , data = heart_data)# --- 2) Logistic regression modellogit_mod <-glm(heart_disease ~ fast_food_spend# + coffee_drinker , data = heart_data, family ="binomial")
Compute alternative models’ predictions
We can now extract the coefficients from the linear regression and logistic models, then estimate the predicted outcomes from these models: lin_pred, logit_pred, and logistic_pred (i.e. the conversion form log(odds) to probability).
# --- 1) Extract coefficients from linear regression modelintercept_lin <-coef(linear_mod)[1]fast_food_spend_lin <-coef(linear_mod)[2]coffee_drinker_lin <-coef(linear_mod)[3]# --- 2) Extract coefficients from logit regression modelintercept_logit <-coef(logit_mod)[1]fast_food_spend_logit <-coef(logit_mod)[2]coffee_drinker_logit <-coef(logit_mod)[3]# --- Estimate predicted data from different models heart_data <- heart_data %>%mutate(# Convert outcome variable to factorheart_disease_factor =factor(heart_disease, labels =c("No Disease (Y=0)", "Disease (Y=1)")),# 1) Linear model predictionlin_pred = intercept_lin + fast_food_spend_lin * fast_food_spend, # 2) Logit model predictionlogit_pred = intercept_logit + fast_food_spend_logit * fast_food_spend, coffee_drinker,# 3) Convert logit to probability (logistic model prediction)logistic_pred =1/ (1+exp(-logit_pred)) ) %>%arrange(fast_food_spend)
Plot alternative models’ outcomes
We can also plot the predicted outcomes from the 3 models to see how they differ.
# --- Plot ggplot(heart_data, aes(x = fast_food_spend)) +# Actual dataset observations (Y=0, Y=1 ) using `color =`geom_jitter(aes(y = heart_disease, color = heart_disease_factor), width =200, height =0.03, alpha =0.75, size =2, shape =16) +# Models' predictions (smooth trends)geom_smooth(aes(y = lin_pred, color ="Linear Regression"), method ="lm", se =FALSE, linewidth =1.25, linetype ="dashed") +geom_smooth(aes(y = logit_pred, color ="Logit (Log-Odds)"), method ="lm", se =FALSE, linewidth =1.25, linetype ="dotdash") +geom_smooth(aes(y = logistic_pred, color ="Logistic Regression"), method ="glm", method.args =list(family ="binomial"), se =FALSE, linewidth =1.25, linetype ="solid") +# Separate legends: color for dots, color for linesscale_color_manual(name ="Actual Y values & Prediction Models", values =c("No Disease (Y=0)"="#A6A6A6", "Disease (Y=1)"="#4c4c4c","Linear Regression"="#d02e4c","Logit (Log-Odds)"="#239b85", "Logistic Regression"="#BD8723")) +# Define scales for the axesscale_x_continuous(breaks =seq(0, 6500, by =500), limits =c(0, 6500), expand =c(0, 0))+scale_y_continuous(breaks =seq(-3, 3, by = .25)) +coord_cartesian(ylim =c(-1.25,1.25), xlim =c(0, 6500)) +theme_minimal() +labs(title ="Comparing Linear and Logistic Regression Predictions v. actual Y values",#subtitle = "(For simplicity, only fast food spend is considered)",y ="Y = Heart disease [0,1]", x ="Fast food spend [US$/yr]", color ="Actual Y values and Predictions")
Plot alternative models’ outcomes
+ (The actual data points are shown as the grey dots) + The linear model predicts values that are ≠ 0 and 1, which poorly fit the actual data + The logit model predicts log(odds) ranging from -Inf to +Inf, which is not interpretable + The logistic model squeezesprobabilities between 0 and 1, which fits the data better
Linear regression didn’t work!
Besides the poor fit, recall linear regression models implies certain assumptions:
Linear relationship between Y and the predictors X1 and X2
Residuals must be 1) approximately normally distributed and 2) uncorrelated
Homoscedasticity: residuals should have constant variance
Non collinearity: predictor variables should not be highly correlated with each other
Figure 1: Diagnostic plots for a hypothetical linear regression model 👎🏻
Fitting a logistic regression model
Now that we are convinced…
…let’s fit instead a logistic regression model to the heart_train data using:
the glm function for Generalized Linear Models,
with argument family = binomial to specify logistic regression which will use logit as the link function,
here we employ all the 3 predictor variables in the dataset (after ~).
# Fit a logistic regression modelheart_model <-glm(heart_disease ~ coffee_drinker + fast_food_spend + income,data = heart_train, family =binomial(link ="logit"))
Model summary and coefficients
Table 1 (next) shows the model output, with the coefficient estimate for each predictor.
The broom::tidy function converts the model summary into a more readable data frame.
The odds ratio (= exponentiated coefficient estimate) is more interpretable than the coefficient itself.
# Convert model's output summary into data frameheart_model_coef <- broom::tidy(heart_model) %>%# improve readability of significance levels dplyr::mutate('signif. lev.'=case_when(`p.value`<0.001~"***",`p.value`<0.01~"**",`p.value`<0.05~"*",TRUE~""))%>%# add odds ratio column dplyr::mutate(odds_ratio =exp(estimate)) %>% dplyr::relocate(odds_ratio, .after = estimate) %>% dplyr::mutate(across(where(is.numeric), ~round(.x, 4))) %>%# format as table knitr::kable() %>%# reduce font sizekable_styling(font_size =20) %>%# add table title kableExtra::add_header_above(c("Logistic Regression Analysis of Heart Disease Risk Factors"=7))heart_model_coef
Model summary and coefficients
Table 1: Logistic regression model output
+ estimates of coefficients are in the form of natural logarithm of the odds (log (odds)) of the event happening (Heart Disease) + a positive estimate indicates an increase in the odds of having Heart Desease + a negative estimate indicates a decrease in the odds of having Heart Desease + odds ratio = the exponentiated coefficient estimate
Logistic Regression Analysis of Heart Disease Risk Factors
term
estimate
odds_ratio
std.error
statistic
p.value
signif. lev.
(Intercept)
-11.0554
0.0000
0.6040
-18.3051
0.0000
***
coffee_drinkerYes_Coffee
-0.7296
0.4821
0.2910
-2.5071
0.0122
*
fast_food_spend
0.0024
1.0024
0.0001
20.6018
0.0000
***
income
0.0000
1.0000
0.0000
-0.2299
0.8182
Interpreting the logistic coefficients
term
estimate
odds_ratio
std.error
statistic
p.value
signif. lev.
(Intercept)
-11.0554
0.0000
0.6040
-18.3051
0.0000
***
coffee_drinkerYes_Coffee
-0.7296
0.4821
0.2910
-2.5071
0.0122
*
fast_food_spend
0.0024
1.0024
0.0001
20.6018
0.0000
***
income
0.0000
1.0000
0.0000
-0.2299
0.8182
Intercept: gives the log-odds of heart disease when all predictor variables are zero. This is not generally interpreted, but the highly negative value suggests very low probability of heart disease in the sample of reference.
(If interpreted) the intercept term -11.0554 would translate as probability\(P = e^{-11.0554} / (1 + e^{-11.0554}) = 0.00002\) or \(0.002\%\).
…which means: when \(\text{coffee_drinker} = NO\), and \(\text{fast_food_spend} = 0\), and \(\text{income} = 0\), the probability of heart disease is as low as \(0.002\%\).
Income: Based on a p-value = 0.8182, we conclude that income is not significantly associated with heart disease. (Anyhow, the odds ratio of ≈1 would suggest no change in odds based on income.)
The coefficient of fast food $$ 🍔🍟
term
estimate
odds_ratio
std.error
statistic
p.value
signif. lev.
(Intercept)
-11.0554
0.0000
0.6040
-18.3051
0
***
fast_food_spend
0.0024
1.0024
0.0001
20.6018
0
***
The positive coefficient of Fast Food Annual Spending (US$), 0.0024 (highly statistically significant as p-value = 0.0000), suggests a positive association with heart disease.
This means that for each additional\(\Delta X_1 = +1 \; US\$\) spent on fast food annually:
the log(odds) of heart disease increases by: \(\Delta \log(\text{odds}) = \beta_{1} \times \Delta X_1 = 0.0024 \times 1 = 0.0024\)
the odds of heart disease (with spending) increase by a factor of: \(OR = e^{0.0024} \approx 1.0024\) (compared to without spending).
The probability of heart disease is computed using the logistic function: \(P_{HD=1} = \frac{e^{\beta_0 + (\beta_1 \times X_1)}}{1 + e^{\beta_0 + (\beta_1 \times X_1)}}\)
Solving for \(\beta_0 = -11.0554\), \(\beta_1 = 0.0024\) and \(X_1 = 1\) gives:
which indicates a probability of heart disease is as low as \(0.00159\%\) when fast_food_spend\(= 1\; US\$\).
The coefficient of fast food $$ 🍔🍟
term
estimate
odds_ratio
std.error
statistic
p.value
signif. lev.
(Intercept)
-11.0554
0.0000
0.6040
-18.3051
0
***
fast_food_spend
0.0024
1.0024
0.0001
20.6018
0
***
However, considering that this is annual spending, we should use a more adequate scale!
For example, for an additional\(\Delta X_1 = +100 \; US\$\) spent on fast food annually:
the log(odds) of heart disease increases by: \(\Delta \log(\text{odds}) = \beta_{1} \times \Delta X_1 = 0.0024 \times 100 = 0.24\)
the odds of heart disease (with spending) increase by a factor of: \(OR = e^{0.24} \approx 1.271\) (compared to without spending).
The probability of heart disease is computed using the logistic function: \(P_{HD=1} = \frac{e^{\beta_0 + (\beta_1 \times X_1)}}{1 + e^{\beta_0 + (\beta_1 \times X_1)}}\)
Solving for \(\beta_0 = -11.0554\), \(\beta_1 = 0.0024\) and and \(X_1 = 100\) gives: \[P_{HD=1} = \frac{e^{-11.0554 + (0.0024 \times 100)}}{1 + e^{-11.0554 + (0.0024 \times 100)}} \approx 0.00002008\]
which (this time) indicates a (still very low probability) of heart disease at \(0.002008\%\) when fast_food_spend\(= 100\; US\$\).
The coefficient of coffee drinking ☕️
term
estimate
odds_ratio
std.error
statistic
p.value
signif. lev.
(Intercept)
-11.0554
0.0000
0.604
-18.3051
0.0000
***
coffee_drinkerYes_Coffee
-0.7296
0.4821
0.291
-2.5071
0.0122
*
The negative coefficient of Coffee Drinker (=YES), -0.7296 expressed in log-odds, means that coffee drinking is associated with lower odds of having heart disease.
Transforming the estimate into odds ratio, we obtain \(0.48 = 48\%\), which tells that coffee-drinkers have 0.48 (or 48%) lower odds than non-coffee drinkers to experience heart disease (holding all other predictors fixed). \[ \text{Odds Ratio} = e^{-0.7296} = 0.48 \]
Alternatively, we could say that non-coffee drinkers have \(1-0.48 = 0.52 = 52\%\) higher odds of having heart disease compared to coffee drinkers.
This effect is statistically significant as p-value = 0.0122.
Wait, is drinking coffee good or bad? 🤔
Our plot above showed that there was a higher proportion of coffee drinkers with heart disease as compared to non coffee drinkers. However, our model just told us that coffee drinking is associated with a decrease in the likelihood of having heart disease.
How can that be❓
It’s because coffee_drinking and fast__food_spend are correlated so, on it’s own, it would appear as if coffee drinking were associated with heart disease, but this is only because coffee drinking is also associated with fast food spend, which our model tells us is the real contributor to heart disease.
🖍️🖍️ qui
Making predictions from logistic regression model
# Make predictionsheart_train$heart_disease_pred <-predict(heart_model, type ="response")
Converting predictions into classifications
# Convert predictions to classificationsheart_train$heart_disease_pred_class <-ifelse(heart_train$heart_disease_pred >0.5, 1, 0)
Evaluating the model]
ROC curve]
# Load the pROC packagelibrary(pROC)# Create a ROC curveroc_curve <-roc(heart_train$heart_disease, heart_train$heart_disease_pred)# Plot the ROC curveplot(roc_curve, col ="blue", main ="ROC Curve", legacy.axes =TRUE)
Name: Biopsy Data on Breast Cancer Patients Documentation: See reference on the data downloaded and conditioned for R here https://cran.r-project.org/web/packages/MASS/MASS.pdf Sampling details: This breast cancer database was obtained from the University of Wisconsin Hospitals, Madison from Dr. William H. Wolberg. He assessed biopsies of breast tumours for 699 patients up to 15 July 1992; each of nine attributes has been scored on a scale of 1 to 10, and the outcome is also known. The dataset contains the original Wisconsin breast cancer data with 699 observations on 11 variables.
Importing Dataset biopsy]
The data can be interactively obtained form the MASS R package
# (after loading pckg)# library(MASS) # I can call utils::data(biopsy)
biopsy variables with description]
Variable
Type
Description
id
character
Sample id
V1
integer 1 - 10
clump thickness
V2
integer 1 - 10
uniformity of cell size
V3
integer 1 - 10
uniformity of cell shape
V4
integer 1 - 10
marginal adhesion
V5
integer 1 - 10
single epithelial cell size
V6
integer 1 - 10
bare nuclei (16 values are missing)
V7
integer 1 - 10
bland chromatin
V8
integer 1 - 10
normal nucleoli
V9
integer 1 - 10
mitoses
class
factor
benign or malignant
biopsy variables exploration]
The biopsy data contains 699 observations of 9 continuous variables, V1, V2, …, V9.
The dataset also contains a character variable: id, and a factor variable: class, with two levels (“benign” and “malignant”).
There is one incomplete variable V6 = “bare nuclei” with 16 missing values.
remember the package skimr for exploring a dataframe?
# check if vars have missing valuesbiopsy %>%# select only variables starting with "V" skimr::skim(starts_with("V")) %>% dplyr::select(skim_variable, n_missing)
We can decide what to do in these cases (informed by our knowledge of the dataset):
Option 1) We drop the observation with incomplete data (i.e. with missing values for V6 = “bare nuclei”) with 16 missing values.
# remove rows with missing valuesbiopsy_drop <- biopsy %>% dplyr::filter(!is.na(V6))mean(biopsy_drop$V6)
[1] 3.544656
Option 2) We impute the missing values with the mean of the variable V6 = “bare nuclei”.
# impute missing values with the median of the variablebiopsy_impute <- biopsy %>% dplyr::mutate(V6 =ifelse(is.na(V6), median(V6, na.rm =TRUE), V6))mean(biopsy_impute$V6)
[1] 3.486409
biopsy dataset exploration]
Biopsied cells of 700 breast cancer tumors, used to determine if the tumors were benign or malignant.
This determination was based on 9 characteristics of the cells, ranked from 1(benign) to 10(malignant):
1) Clump Thickness – How the cells aggregate. If monolayered they are benign and if clumped on top of each other they are malignant
2) Uniform Size – All cells of the same type should be the same size.
3) Uniform Shape If cells vary in cell shape they could be malignant
4) Marginal Adhesion – Healthy cells have a strong ability to stick together whereas cancerous cells do not
5) Single Epithelial Size – If epithelial cells are not equal in size, it could be a sign of cancer
6) Bare nuclei – If the nucleus of the cell is not surrounded by cytoplasm, the cell could be malignant
7) Bland Chromatin – If the chromatin’s texture is coarse the cell could be malignant
8) Normal Nucleoli – In a healthy cell the nucleoli is small and hard detect via imagery. Enlarged nucleoli could be a sign of cancer
9) Mitosis – cells that multiply at an uncontrollable rate could be malignant
biopsy dataset preparation]
The explanatory variable(s) (clump_thickness, …, mitosis) can be renamed for better readability
The observations with missing values (bare_nuclei) are removed for simplicity
Patient ID (id) can be dropped as it is not used in this analysis
# Create a clean version of the datasetbiopsy_clean <- biopsy %>%# rename the columns (new = old)rename(id = ID,clump_thickness = V1,uniform_size = V2,uniform_shape = V3,marginal_adhesion = V4,single_epith_size = V5,bare_nuclei = V6,bland_chromatin = V7,normal_nuclei = V8,mitosis = V9,class = class) %>%# remove rows with missing valuesna.omit(bare_nuclei) %>%# remove the id columnselect(-id)# check the structure of the datasetpaint::paint(biopsy_clean)
data.frame [683, 10]
clump_thickness int 5 5 3 6 4 8
uniform_size int 1 4 1 8 1 10
uniform_shape int 1 4 1 8 1 10
marginal_adhesion int 1 5 1 1 3 8
single_epith_size int 2 7 2 3 2 7
bare_nuclei int 1 10 2 4 1 10
bland_chromatin int 3 3 3 3 3 9
normal_nuclei int 1 2 1 7 1 7
mitosis int 1 1 1 1 1 1
class fct benign benign benign benign benign ma~
biopsy sample splitting]
In Machine Learning, it is good practice to split the data into training and testing sets.
We will use the training set (80%) to fit the model and then the testing set (20%) to evaluate the model’s performance.
set.seed(123)# Obtain 2 sub-samples from the dataset: training and testingsample <-sample(c(TRUE, FALSE), nrow(biopsy_clean), replace =TRUE , prob =c(0.8, 0.2) )biopsy_train <- biopsy_clean[sample,]biopsy_test <- biopsy_clean[!sample,]
Which results in:
# check the structure of the resulting datasetsdim(biopsy_train)
[1] 542 10
dim(biopsy_test)
[1] 141 10
Indipendent variables’ visualization]
We create a new df biopsy_train2 (with only 3 columns)
Then, in Figure 2, we visualize the distribution of the explanatory variables, where each is plotted between the two classes of the tumor.
# New df for plottingbiopsy_train2 <-data.frame("level"=c(biopsy_train$clump_thickness, biopsy_train$uniform_size, biopsy_train$uniform_shape, biopsy_train$marginal_adhesion, biopsy_train$single_epith_size, biopsy_train$bare_nuclei, biopsy_train$bland_chromatin, biopsy_train$normal_nuclei, biopsy_train$mitosis),"type"=c("Clump Thickness", "Uniform Size", "Unifrom Shape","Marginal Adhesion", "Single Epithilial Size", "Bare Nuclei","Bland Chromatin", "Normal Nuclei", "Mitosis"), "class"=c(biopsy_train$class))# Plotggplot(biopsy_train2, aes(x = level, y = class , colour = class)) +geom_boxplot(fill =NA) +scale_color_manual(values =c("#005ca1", "#9b2339")) +geom_jitter(aes(fill = class), alpha =0.25, shape =21, width =0.2) +scale_fill_manual(values =c("#57b7ff", "#e07689")) +facet_wrap(~type, scales ="free") +theme(plot.title =element_text(size =13,face="bold", color ="#873c4a"),axis.text.x =element_text(size=12,face="italic"), axis.text.y =element_text(size=12,face="italic"),legend.position ="none") +labs(title ="Distribution of each explanatory variable by tumor class (benign/malignant) in samples") +ylab(label ="") +xlab(label ="")
Indipendent variables’ visualization]
Figure 2: Boxplot of the independent variables - values are consitently higher & more dispersed for malignant tumors - values between 1 and 2 are classified as benign and values greater than 2 are classified as malignant
Logistic regr.: model fitting]
We fit a logistic regression model to the biopsy_train data using:
the glm function with argument family = binomial to specify the logistic regression model;
and with Class ~ . to specify an initial model that uses all the variables as predictors (backward elimination approach).
# Building initial model model = stats::glm(class ~ . , family = binomial, data=biopsy_train)
Table 2 shows the model summary, with the coefficient estimate for each predictor.
the broom::tidy function converts the model summary into a data frame.
Table 2: Complete logistic regression model + coefficients are in the form of natural logarithm of the odds of the event happening + positive estimate indicates an increase in the odds of finding a malignant tumor
term
estimate
std.error
statistic
p.value
signif. lev.
(Intercept)
-9.4169
1.1637
-8.0921
0.0000
***
clump_thickness
0.4984
0.1434
3.4758
0.0005
***
uniform_size
0.0992
0.2356
0.4211
0.6737
uniform_shape
0.2809
0.2655
1.0580
0.2901
marginal_adhesion
0.2688
0.1285
2.0917
0.0365
*
single_epith_size
0.0800
0.1679
0.4765
0.6337
bare_nuclei
0.3446
0.0983
3.5054
0.0005
***
bland_chromatin
0.4083
0.1859
2.1958
0.0281
*
normal_nuclei
0.2196
0.1283
1.7119
0.0869
mitosis
0.4356
0.3513
1.2397
0.2151
Logistic regr.: coefficients’ interpretation
In logistic regression, the coefficients are in the form of the natural logarithm of the odds of the response event happening (i.e. \(Y_i = 1\)):
However, with some algebraic transformation, the logit function can be inverted to obtain the probability of the response event happening as a function of the predictors:
Highlight key logistic model performance metrics using broom::glance() function:
AIC: A measure of model quality; lower values indicate a better fit with fewer parameters.
Null Deviance: A measure of model error; how well the response variable can be predicted by a model with only an intercept term.
Deviance: A measure of model error; how well the response variable can be predicted by a model with predictor variables.
lower values mean the model fits the data better!
broom::glance(model)[, c("AIC", "null.deviance","deviance")] %>%# show only performance metrics knitr::kable()
Table 3: Key logistic model performance metrics
AIC
null.deviance
deviance
111.7472
710.4404
91.74724
Logistic regr.: improving the model]
The model includes all variables, but we could make it more parsimonious by removing variables that are not significant!
We use a statistic called the Akaike Information Criterion (AIC) to compare models.
AIC’s calculation gives a penalty for including additional variables.
The model with the lowest AIC is considered the best.
# For example let's fit a model without the variable `uniform_size`model2 =glm(class~ .-uniform_size, family = binomial, data=biopsy_train)
According to the AIC values, the model2 seems better (AIC is lower).
# Compare the AIC values of the 2 modelstibble(Model =c("model", "model2"), AIC =c(AIC(model), AIC(model2) )) %>%kable()
Model
AIC
model
111.7472
model2
109.9322
Logistic regr.: systematic model selection]
The MASS package’s function stepAIC enables to perform a systematic model selection (by AIC):
The direction argument specifies the direction of the stepwise search.
The trace argument (if set to TRUE) prints out all the steps.
The best_model has removed these variables:
uniform_shape
single_epith_size
normal_nuclei
The best_model has the lowest AIC value (from 100 to 98.5), despite a higher Residual Deviance than the full model (from 80 to 80.6), albeit by a very slight amount.
# Select the best model based on AICbest_model <- MASS::stepAIC(model, direction ="both", trace =FALSE)# Compare the AIC values of full and best modeltibble(Model =c("model", "best_model"), AIC =c(AIC(model), AIC(model2)),Deviance =c(deviance(model), deviance(best_model))) %>%kable()
Logistic regr.: systematic model selection]
Model
AIC
Deviance
model
111.7472
91.74724
best_model
109.9322
92.22712
Logistic regr.: predicting on test data]
We can use the predict function to predict the class of the biopsy_test using the best_model.
biopsy_test_pred contains the probability that each of observation (in test data) is malignant.
The classification of the PredictedValue... can be done using different probability thresholds (0.5, 0.4, 0.3, etc.). which will affect the true and false positivity rates.:
PredictedValue_05 is the standard threshold of 0.5.
PredictedValue_04 is a more conservative threshold of 0.4.
PredictedValue_07 is a more aggressive threshold of 0.7.
# Fitted value for the test data 205 samples based on modelbiopsy_test_pred <-predict(best_model, newdata = biopsy_test, type ="response")# Convert the predicted probabilities into 2 predicted classesActualValue <- biopsy_test$class# Different possible thresholds for the predicted probabilitiesPredictedValue_05 <-if_else(biopsy_test_pred >0.5, "pred_malignant", "pred_benign")PredictedValue_04 <-if_else(biopsy_test_pred >0.4, "pred_malignant", "pred_benign")PredictedValue_07 <-if_else(biopsy_test_pred >0.7, "pred_malignant", "pred_benign")
Logistic regr. predictions: confusion matrix]
In diagnosing malignant tumors, it is important to keep the false negative rate low as this would be telling someone who has a malignant tumor that it is benign.
At \(p = 0.7\), the model will predict more false positives than at \(p = 0.4\) (6, v. 4 FN) – which we DON’T WANT
in this situation \(p = 0.4\) is preferable.
Then, we can evaluate the model’s performance on the test data by building a confusion matrix
# Build the confusion matrix with p = 0.4table(ActualValue=biopsy_test$class, PredictedValue_04) %>% knitr::kable()
pred_benign
pred_malignant
benign
97
2
malignant
1
41
# Build the confusion matrix with p = 0.7table(ActualValue=biopsy_test$class, PredictedValue_07) %>% knitr::kable()
pred_benign
pred_malignant
benign
98
1
malignant
2
40
Logistic regr. predictions: accuracy]
We found that a cutoff of 0.4 gives a good balance of low false negatives while still maintaining a high true positive rate.
With the chosen threshold of \(p = 0.4\), we can calculate the model’s accuracy on the test data.
# Build the confusion matrix with p = 0.4conf_matr_04 <-table(ActualValue=biopsy_test$class, PredictedValue_04) # Calculate the accuracyaccuracy <-sum(diag(conf_matr_04)) /sum(conf_matr_04)accuracy
predicted.data <-data.frame(prob.of.malig=biopsy_test_pred, malig = biopsy_test$class)predicted.data <- predicted.data[order(predicted.data$prob.of.malig, decreasing = F),]predicted.data$rank <-1:nrow(predicted.data)plot_ROC <-ggplot(data=predicted.data, aes(x=rank, y=prob.of.malig)) +geom_point(aes(color=malig), alpha=1, shape=4, stroke=2) +xlab("Index") +ylab("Predicted Probability of Tumor Being Malignant") plot_ROC
This shows why lowering the cutoff improves the accuracy of the model as some malignant tumors are being underestimated which would cause false negatives.
Logistic regr.: conclusions]
Uniform Size and Single Epithithial Size were not significant in predicting the malignancy of tumor cells so our model does not include these variables.
Our fitted model reduces the null deviance and AIC without impacting the residual deviance by a significant amount and is able to predict the testing dataset with >90% accuracy.
For further analysis, we could run the model multiple times because our original and revised model are similar.
New training and testing data would help confirm our results and help identify possible overfitting.
🟠 K-MEANS CLUSTERING: EXAMPLE of UNSUPERVISED ML ALGORITHM
…]
PCA: EXAMPLE of UNSUPERVISED ML ALGORITHM
Reducing high-dimensional data to a lower number of variables
biopsy dataset manipulation]
We will:
exclude the non-numerical variables (id and class) before conducting the PCA.
exclude the individuals with missing values using the na.omit() or filter(complete.cases() functions.
We can do both in 2 equivalent ways:
with base R (more compact)
# new (manipulated) dataset data_biopsy <-na.omit(biopsy[,-c(1,11)])
with dplyr (more explicit)
# new (manipulated) dataset data_biopsy <- biopsy %>%# drop incomplete & non-integer columns dplyr::select(-ID, -class) %>%# drop incomplete observations (rows) dplyr::filter(complete.cases(.))
biopsy dataset manipulation]
We obtained a new dataset with 9 variables and 683 observations (instead of the original 699).
The first step of PCA is to calculate the principal components. To accomplish this, we use the prcomp() function from the stats package.
With argument “scale = TRUE” each variable in the biopsy data is scaled to have a mean of 0 and a standard deviation of 1 before calculating the principal components (just like option Autoscaling in MetaboAnalyst)
# calculate principal componentbiopsy_pca <-prcomp(data_biopsy, # standardize variablesscale =TRUE)
Analyze Principal Components
Let’s check out the elements of our obtained biopsy_pca object
(All accessible via the $ operator)
names(biopsy_pca)
[1] "sdev" "rotation" "center" "scale" "x"
“sdev” = the standard deviation of the principal components
“sdev”^2 = the variance of the principal components (eigenvalues of the covariance/correlation matrix)
“rotation” = the matrix of variable loadings (i.e., a matrix whose columns contain the eigenvectors).
“center” and “scale” = the means and standard deviations of the original variables before the transformation;
“x” = the principal component scores (after PCA the observations are expressed in principal component scores)
Analyze Principal Components (cont.)
We can see the summary of the analysis using the summary() function
The first row gives the Standard deviation of each component, which can also be retrieved via biopsy_pca$sdev.
The second row shows the Proportion of Variance, i.e. the percentage of explained variance.
The output suggests the 1st principal component explains around 65% of the total variance, the 2nd principal component explains about 9% of the variance, and this goes on with diminishing proportion for each component.
Cumulative Proportion of variance for components]
The last row from the summary(biopsy_pca), shows the Cumulative Proportion of variance, which calculates the cumulative sum of the Proportion of Variance.
# Extracting Cumulative Proportion from summarysummary(biopsy_pca)$importance[3,]
Once you computed the PCA in R you must decide the number of components to retain based on the obtained results.
VISUALIZING PCA OUTPUTS
Scree plot
There are several ways to decide on the number of components to retain.
One helpful option is visualizing the percentage of explained variance per principal component via a scree plot.
Plotting with the fviz_eig() function from the factoextra package
# Scree plot shows the variance of each principal component factoextra::fviz_eig(biopsy_pca, addlabels =TRUE, ylim =c(0, 70))
Visualization is essential in the interpretation of PCA results. Based on the number of retained principal components, which is usually the first few, the observations expressed in component scores can be plotted in several ways.
Scree plot
The obtained scree plot simply visualizes the output of summary(biopsy_pca).
Principal Component Scores]
After a PCA, the observations are expressed as principal component scores.
We can retrieve the principal component scores for each Variable by calling biopsy_pca$x, and store them in a new dataframe PC_scores.
Next we draw a scatterplot of the observations – expressed in terms of principal components
# Create new object with PC_scoresPC_scores <-as.data.frame(biopsy_pca$x)head(PC_scores)
It is also important to visualize the observations along the new axes (principal components) to interpret the relations in the dataset:
Principal Component Scores plot (adding label variable)]
When data includes a factor variable, like in our case, it may be interesting to show the grouping on the plot as well.
In such cases, the label variable class can be added to the PC set as follows.
# retrieve class variablebiopsy_no_na <-na.omit(biopsy)# adding class grouping variable to PC_scoresPC_scores$Label <- biopsy_no_na$class
The visualization of the observation points (point cloud) could be in 2D or 3D.
Principal Component Scores plot (2D)]
The Scores Plot can be visualized via the ggplot2 package.
grouping is indicated by argument the color = Label;
geom_point() is used for the point cloud.
ggplot(PC_scores, aes(x = PC1, y = PC2, color = Label)) +geom_point() +scale_color_manual(values=c("#245048", "#CC0066")) +ggtitle("Figure 1: Scores Plot") +theme_bw()
Principal Component Scores plot (2D)]
Figure 1 shows the observations projected into the new data space made up of principal components
Principal Component Scores (2D Ellipse Plot)]
Confidence ellipses can also be added to a grouped scatter plot visualized after a PCA. We use the ggplot2 package.
grouping is indicated by argument the color = Label;
geom_point() is used for the point cloud;
the stat_ellipse() function is called to add the ellipses per biopsy group.
ggplot(PC_scores, aes(x = PC1, y = PC2, color = Label)) +geom_point() +scale_color_manual(values=c("#245048", "#CC0066")) +stat_ellipse() +ggtitle("Figure 2: Ellipse Plot") +theme_bw()
Principal Component Scores (2D Ellipse Plot)]
Figure 2 shows the observations projected into the new data space made up of principal components, with 95% confidence regions displayed.
Principal Component Scores plot (3D)]
A 3D scatterplot of observations shows the first 3 principal components’ scores.
For this one, we need the scatterplot3d() function of the scatterplot3d package;
The color argument assigned to the Label variable;
To add a legend, we use the legend() function and specify its coordinates via the xyz.convert() function.
# 3D scatterplot ...plot_3d <-with(PC_scores, scatterplot3d::scatterplot3d(PC_scores$PC1, PC_scores$PC2, PC_scores$PC3, color =as.numeric(Label), pch =19, main ="Figure 3: 3D Scatter Plot", xlab="PC1",ylab="PC2",zlab="PC3"))# ... + legendlegend(plot_3d$xyz.convert(0.5, 0.7, 0.5), pch =19, yjust=-0.6,xjust=-0.9,legend =levels(PC_scores$Label), col =seq_along(levels(PC_scores$Label)))
Principal Component Scores plot (3D)]
Figure 3 shows the observations projected into the new 3D data space made up of principal components.
Biplot: principal components v. original variables]
Next, we create another special type of scatterplot (a biplot) to understand the relationship between the principal components and the original variables.
In the biplot each of the observations is projected onto a scatterplot that uses the first and second principal components as the axes.
For this plot, we use the fviz_pca_biplot() function from the factoextra package
We will specify the color for the variables, or rather, for the “loading vectors”
The habillage argument allows to highlight with color the grouping by class
Biplot: principal components v. original variables]
The axes show the principal component scores, and the vectors are the loading vectors
Interpreting biplot output
Biplots have two key elements: scores (the 2 axes) and loadings (the vectors). As in the scores plot, each point represents an observation projected in the space of principal components where:
Biopsies of the same class are located closer to each other, which indicates that they have similar scores referred to the 2 main principal components;
The loading vectors show strength and direction of association of original variables with new PC variables.
As expected from PCA, the single PC1 accounts for variance in almost all original variables, while V9 has the major projection along PC2.
Interpreting biplot output (cont.)
scores <- biopsy_pca$xloadings <- biopsy_pca$rotation# excerpt of first 2 componentsloadings[ ,1:2]
Motivated the choice of learning/using R for scientific quantitative analysis, and lay out some fundamental concepts in biostatistics with concrete R coding examples.
Consolidated understanding of inferential statistic, through R coding examples conducted on real biostatistics research data.
Discussed the relationship between any two variables, and introduce a widely used analytical tool: regression.
Presented a popular ML technique for dimensionality reduction (PCA), performed both with MetaboAnalyst and R.
Introduction to power analysis to define the correct sample size for hypotheses testing and discussion of how ML approaches deal with available data.
Final thoughts
While the workshop only allowed for a synthetic overview of fundamental ideas, it hopefully provided a solid foundation on the most common statistical analysis you will likely run in your daily work:
Thorough understanding of the input data and the data collection process
Univariate and bivariate exploratory analysis (accompanied by visual intuition) to form hypothesis
Upon verifying the assumptions, we fit data to hypothesized model(s)
Assessment of the model performance (\(R^2\), \(Adj. R^2\), \(F-Statistic\), etc.)
You should now have a solid grasp on the R language to keep using and exploring the huge potential of this programming ecosystem
We only scratched the surface in terms of ML classification and prediction models, but we got a hang of the fundamental steps and some useful tools that might serve us also in more advanced analysis